Variable frequency electrostatic drive

ABSTRACT

A variable speed drive for an electrostatic motor provides feedback control by conversion of measured current phases provided to the motor into a vector in a rotating rotor framework. This vector is used for evaluating corrective voltages and then reconverted to a non-rotating framework for application to the motor electrodes. Current-source drive circuits provide current stabilized outputs making such sophisticated control tractable.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under 1452230 awarded bythe National Science Foundation. The government has certain rights inthe invention.

CROSS-REFERENCE TO RELATED APPLICATIONS

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BACKGROUND OF THE INVENTION

The present invention relates to electrostatic motors and in particularto a drive providing electrical power to an electrostatic motor.

Electrostatic motors operate by exploiting forces generated byelectrical fields on a respective stator and rotor, for example, asestablished on opposed plates on the stator and rotor. Electrostaticmotors may be distinguished from conventional electromagnetic motorswhich exploit forces generated by interacting magnetic fields generatedby the stator and rotor, for example, generated by current-carryingelectrical coils or a permanent magnet.

Synchronous electrostatic motors are a class of electrostatic motorsthat provide a rotor presenting a substantially static electricalpolarization which causes the rotor to rotate synchronously with arotating electrostatic field generated by the stator. The statorgenerates this rotating electrostatic field by applying different phasesof a periodic voltage to a set of circumferentially or axially displacedelectrodes.

Electrical power to the stator, producing the desired rotatingelectrostatic field, may be generated, for example, by means of ahigh-voltage, three-phase, sinusoidal power from a generator applied toappropriately phased stator electrodes. Electrical power may also besynthesized by a step approximation or by pulse width modulationsynthesis using high-voltage transistors to switch different polaritiesof electrical voltage among the stator electrodes. Current methods ofcontrolling electrostatic motors provide relatively primitive control oftorque and speed resulting in unnecessary power consumption orunutilized capability.

SUMMARY OF THE INVENTION

The present inventors have developed an electrical drive for anelectrostatic motor that can tailor the stator voltages precisely to therotor position to provide greatly enhanced electrostatic motor control.Current-source drivers are used to adapt the field controltransformation techniques of electromagnetic motors to the electrostaticdomain. The novel saliency of an electrostatic motor is modeled toprovide for improved characterization of the necessary current vectorangles for efficient production of torque. This control strategyutilizes a reference frame transformation to continuously monitor andposition the relative magnitudes and angles of electrical quantities forprecise torque modulation and motion control. Prior techniques forelectrostatic motors were unable to extract maximum torque per volt orfield oriented control in dynamic variable speed situations.

Specifically, one embodiment of the invention provides a variable speeddrive for an electrostatic motor. The variable speed drive includes aset of current-source drives adapted to connect to the multiple fieldelectrodes of the electrostatic motor and a rotor position detector. Ad-q transformation circuit receives a position signal from the rotorposition detector and measures of the outputs of the independent currentdrives to develop a measured d-q vectors (being a transformation to therotating stator framework described below), and an input receives thedesired d-q vectors. A comparison circuit provides an error vector basedon the measured d-q vectors and desired d-q vectors. The comparisoncircuit may scale the d-q error vector by a regulator gain and thisscaled d-q error is then provided to an ABC transformation circuit(reversing the d-q transformation) receiving the position signal andproducing an output based on the error vector to develop a set ofelectrode signals provided to the current-source drives for driving thestator electrodes.

It is thus a feature of at least one embodiment of the invention toprovide for sophisticated control of electrostatic motors, for example,including torque and speed control, by implementing field controlmethodology into the electrostatic framework. Current-source drives,capable of modulating ampere-seconds to the electrostatic machine basedon a received command, permits tractable implementation of d-q controlwith simplified feedback loops.

The current-source drives may provide a set of electrical switches inseries with a current-source implemented by an inductance on the DC sideserving to provide a stiff current that may be modulated into the statorelectrodes.

It is thus a feature of at least one embodiment of the invention toprovide a robust current-source drive using the currentregulating/energy storage properties of inductances on the AC side tomanage current demand present in operating electrostatic motors.

The current-source drives may include a multi-phase H-bridge ofelectrical switches receiving current from an inductance operating toregulate current flow.

It is thus a feature of at least one embodiment of the invention to makeuse of conventional bridge circuitry used for electrical control toprovide a flexible synthesis of arbitrary waveforms in thecurrent-source drives.

The current regulating inductance may be part of a multiphasetransformer on the AC side and the multiphase H-bridge of electricalswitches may provide current to a multiphase transformer.

It is thus a feature of at least one embodiment of the invention toobtain the needed current-regulating inductance as part of a transformerthat can step up voltage presented to the electrostatic motor allowingthe semiconductor devices of the H-bridge to operate at preferred lowervoltages.

The current-source drive may provide for a controller controllingswitching the electrical switches of the multiphase H-bridge accordingto a sensed current output and sensed voltage output of the multiphasetransformer.

It is thus a feature of at least one embodiment of the invention topermit the simultaneous use of current and voltage feedback to implementresponsive current-source drives needed for variable speed control in acircuit having a transformer interposed between the H-bridge and theelectrostatic motor.

The variable speed drive may include an input circuit generating thedesired d-q vector based on a received command selected from thecommands of torque and speed to control current applied to the statorelectrodes.

It is thus a feature of at least one embodiment of the invention toprovide a general system that may receive voltage, current, position,torque, speed or other types of commands by converting those commandsinto a vector in d-q space.

The input circuit may select an angle for the desired d-q vector betweenthe d- and q-motor axes as a function of coupling capacitance between arotor electrode and a stator electrode and a coupling capacitancebetween two stator electrodes.

It is thus a feature of at least one embodiment of the invention toprovide a modeling of a salient pole electrostatic motor that considersinter-capacitive coupling of the motor components affecting an idealphase of the stator voltage vector with respect to rotor position.

The input circuit selects an angle for the desired d-q voltage vectorbetween the d- and q-motor axes according to the formula:

$\gamma_{{ma}\; x} = {{{- a}\;{\sin\left( \frac{{C_{1{mf}}V_{fd}} \pm \sqrt{{C_{1{mf}}^{2}V_{fd}^{2}} + {8C_{2\; s}^{2}V_{s}^{2}}}}{4C_{2s}V_{s}} \right)}} \pm {5{\%.}}}$

where:

-   -   γ_(max) is an optimum angle between the desired d-q voltage        vector and the q-axis, normally a constant value for        steady-state motor operation;    -   C_(1mf) is the magnitude of the fundamental of the coupling        capacitance between any one of the stator terminals (a, b, c) to        any one of the rotor terminals (Vf+ or Vf−);    -   C_(2s) is the magnitude of the second harmonic of the coupling        capacitance between any two stator terminals (a-b, b-c, c-a);    -   V_(fd) is the magnitude of the do rotor voltage; and    -   V_(s) is the magnitude of the fundamental of the stator voltage.

It is thus a feature of at least one embodiment of the invention toprovide a drive that can select an optimal phase angle of the appliedstator voltages for maximum torque per volt in a motor with field andsaliency torque.

Alternatively, the value of γ_(max) may be selected from the groupconsisting of 0 and π (each ±5%) for a motor with field torque only andπ/4, −π/4, 3π/4, −3π/4 (each ±5%) for a motor with saliency torque only.

These particular features and advantages may apply to only someembodiments falling within the claims and thus do not define the scopeof the invention.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a simplified exploded diagram of the drive system of thepresent invention used with a “nested cylinder” type motor design;

FIG. 2 is a figure similar to FIG. 1 of the drive system of the presentinvention used with an “opposed plate” type design;

FIG. 3 is a detailed block diagram of the principal control elements ofthe drive of FIGS. 1 and 2 of the outputs connected to electrodes of asimplified three-phase electrostatic motor providing current-sourceoutputs for each electrode;

FIG. 4 is a first embodiment of the current-source drive of FIG. 3employing a single current regulating inductance;

FIG. 5 is a second embodiment of the current-source drive a FIG. 3including three independent current-regulating inductances;

FIG. 6 is a third embodiment of the current-source drive of FIG. 2incorporating a step up transformer for minimizing the drop voltageacross the switching semiconductors and using leakage inductance of thetransformer for current regulation;

FIG. 7 is a plot of torque versus angle of the stator field relative tothe rotor as informed by the present invention's abstraction of torqueinto field and elastance torques showing determination of maximum torqueper volt, this plot positioned adjacent to a vector diagram depictingthe angle of the stator field in a d-q reference frame; and

FIG. 8 is a diagrammatic representation of terms used to determine theangle relationship of FIG. 7

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring now to FIG. 1, an electrostatic drive system 10 may include anelectrostatic motor 12 having a generally cylindrical stator 14supporting internal, longitudinally extending and circumferentiallydisplaced stator electrodes 16. Fitting within the stator 14 is a rotor18 having corresponding, outwardly exposed longitudinally extending andcircumferentially displaced rotor electrodes 20 interacting with thestator electrodes 16 when the rotor 18 is positioned within the stator14.

In an alternative design, shown in FIG. 2, the stator 14 may be one ormore disk-shaped plates having radially extending, circumferentiallydisplaced stator electrodes 16 interacting with corresponding radiallyextending, circumferentially displaced rotor electrodes 20 on acorresponding disk-shaped rotor 18 positioned adjacent to thedisk-shaped stator 14 for interaction therewith.

Motors of this type are described in U.S. Pat. No. 9,184,676 as well asapplications 2016/0211775 and 2016/0344306 all assigned to the assigneeof the present invention and incorporated by reference.

In both of the designs of FIGS. 1 and 2, the rotors 18 may be supportedfor rotation on driveshafts 24 extending axially outward for theextraction of mechanical work. A slip ring or brushless type of powertransfer (capacitive or inductive) system 22 attached to the driveshaft24 allows electricity from a stationary rotor power supply 26 to beconducted to the rotating rotor electrodes 20 through terminals Vf+ andVf− as is generally understood in the art to provide an electrostaticpolarization of the rotor 18.

The electrostatic drive systems 10 provide for position monitoring ofthe rotors 18 with respect to a stationary stator 14 through the use ofa position detection system 30. The position detection system 30 may bea resolver or encoder mechanically attached to the driveshaft 24 toprovide a signal uniquely identifying the location of the rotor 18 withrespect to the stator 14 in a range of 0 to 2 π radians. Alternatively,similar measurements may be provided by other position sensors orestimators well known in the art including, for example, “back MMF”(magneto motive force) sensors which equate changes in the back MMFsensed at the stator electrodes 16 to a rotational position, or“saliency tracking” sensors which inject a high-frequency signal, forexample, into the output of the rotor power supply 26 and detect changesin that signal as coupled to the stator 14 caused by the variations inelectrical coupling between the stator and rotor caused by motorsaliency. Each of these alternatives will be termed a positiondetermination system 30.

In both the designs of FIGS. 1 and 2, a variable speed drive 32 of thepresent invention may provide for controlled application of power to thestator electrodes 16 of the stator 14 based on the position signalgenerated by the position detection system 30. In this regard, thevariable speed drive 32 may receive a command signal 34, for example,speed or torque or other related quantity, and determine the propervariable currents to be applied to the stator electrodes 16 necessary toprovide operation of the electrostatic motor 12 in conformance with thatcommand signal 34. As such, the output of the variable speed drive 32will provide multiple phases 36 (also designated A,B,C, for a threephase embodiment) associated with different stator electrodes 16providing sinusoidal or other continuously varying signals to thosestator electrodes 16 necessary to maximize motor performance.

Referring now to FIG. 3, the instantaneous values of the output phases36 required for a given command signal 34 can be a function of not onlythe command signal 34 but also rotor position and the characteristics ofthe motor 12. This processing necessary to generate the output phases 36may be simplified through a coordinate transformation known inconventional electromagnetic motors in which constantly varying multiplephases 36 are mapped to a reference frame rotating with the motor rotor18. This reference frame is termed the d-q reference frame where the daxis (the direct axis) is aligned with the positive electrode on therotor 20 and the q axis (the quadrature axis) is positioned at 90degrees with respect to the d axis. Viewed in this reference frame, thecomplexity of the waveforms at the multiple phases 36 (termed A, B, C,for an example, in a three-phase system) devolves to a single vectorthat is largely unvarying for steady-state operation of the motor 12.Details of this transformation in the context of electromagneticmachines are described, for example, at D. W. Novotny and T. A. Lipo,“Vector Control and Dynamics of AC Drives,” 1st edition, OxfordUniversity Press, 1996 (including pages 88-102) with the underlyingmathematics also applicable to the present invention.

Using this transformation, the present invention provides a feedbackcontrol of a current-source drive 40 having phases 36 connected to eachof the stator electrodes 16. In this regard, current and in some casesvoltages from each of these phases 36 are measured and thesemeasurements received by ABC-dq transformation circuit 42. The ABC-dqtransformation circuit 42 also receives a position signal 44 from theposition detection system 30 to convert the received phase signals (A,B, C) into a vector in d-q space termed the “measured” d-q vector 48.

The input command signal 34 will be converted to a similar “desired” d-qvector 50 by input conversion circuit 52. This desired d-q vector 50will generally have a different angle and different magnitude than themeasured d-q vector 48 when the electrostatic motor 12 is not operatingin steady-state. When the input command signal 34 is a torque value, themagnitude of the desired d-q vector 50 will be proportional to thedesired torque and the ideal angle with respect to the q-axis willdepend on the type of motor 12. For a non-salient machine, the anglewill simply be zero degrees (a desired d-q vector 50 aligned with theq-axis); however, for a salient machine this calculation will be morecomplex as will be discussed below. The ideal angle is one that providesmaximum torque per voltage thereby reducing motor losses.

Once the desired d-q vector 50 is determined, it is compared it to themeasured d-q vector 48 to produce an update value 53 at comparison block54 which controls the current-source is 40. In the simplest case, updatevalue 53 is simply a difference between the desired d-q vector 50 andthe measured d-q vector 48; however, alternatively, this difference maybe further processed, for example, underproportional/integral/derivative type control strategies in which theupdate value 53 is a weighted combination of the difference value, atime running integration of this difference value, and a derivative ofthis difference value. It will also be appreciated that other controlstrategies may be introduced in comparison at block 54 includingfeedback and/or feedforward of other measured variables derived from themotor 12.

Referring still to FIG. 3, the correction value 53 is then provided toan dq-ABC transformation circuit 56 operating in the reverse directionas the ABC-dq transformation circuit 42 (as inverse transforms) tochange the correction value 53, being a vector in d-q space, phases 36in a nonrotating frame.

This feedback control process, traversing the loop of ABC-dqtransformation circuit 42 and dq-ABC transformation circuit 56,continues during operation of the motor 12.

When the command signal 34 is a different value, for example, a desiredrotational speed (e.g., RPM), an additional, optional feedback loop maybe incorporated, for example, using the position signal 44 to deducespeed and using a difference between the desired RPM of the commandsignal 34 and the deduced RPM, at optional comparison block 58, tocreate a torque value that may then be treated as discussed above withrespect to the torque signal. Other input signals can also be handled inthis manner, and in this regard and the invention contemplates thatprogrammable command signals 34 may be used, for example, for soft startand stop of the motor 12 as well as different regimes at different motorRPMs or operating conditions.

The ABC-dq transformation circuit 42, input conversion circuit 52,comparison block 54, and dq-ABC transformation circuit 56 may beimplemented by discrete circuitry or preferably by a high-speed computerprocessor executing a program stored in non-transient computer memoryfor example as firmware and employing analog-to-digital converters tooperate in a digital domain.

Referring now to FIGS. 4 and 5, practical implication of sophisticatedfield control of an electrostatic motor is enabled by the ability togenerate “stiff” current output signals at the power levels needed todrive electrostatic motor 12, that is, outputs that can provideopen-loop current control in the face of rapidly fluctuating voltages atthe multiple phases 36 caused by changes in capacitive coupling withrotation of the motor 12. The invention contemplates that theelectrostatic motors 12 will operate at powers in excess of 10 watts,typically in excess of 100 watts, and desirably in excess of 1000 watts.

The necessary “current-source” outputs may be produced through the useof one or more series inductive elements exploiting a feature ofinductance that resists changes in the current flowing through theinductor, a feature of the buildup of self-induced energy within themagnetic field of the inductor. The present invention recognizes thatthis property can be enlisted to provide sufficient output currentstability to synthesized current waveforms without unduly preventingdynamic control of that current necessary for “field control” of themotor or variable speed capabilities. In this regard the inductance mustbe of a size to provide current regulation (and hence energy storage) atthe expected motor power levels providing, for example, for the controlof current output to the motor to within 25 percent of the command valuecontrolling the semiconductor switches, and typically within 10 percent,and desirably within five percent.

In a first such implementation, a current regulating inductor 70provides current to a set of solid-state switches 72, for example,transistors such as MOSFET transistors, receiving ABC current valuesfrom the switching logic circuit 73. The solid-state switches 72, forexample, are configured in an H-bridge where each of the phases 36connects to a junction between a pair of series-connected switches 72,the pair in turn spanning a positive power rail 74 and a negative powerrail 76 providing a direct current stabilized by inductor 70.Rudimentary use of this circuit can produce square wave outputs; howeverthe present invention contemplates that the phases 36 produced arecontinuous waveforms of arbitrary shape and frequency dictated by thecontrol algorithm. Accordingly the switches 72 will receive controlsignals determining their switch state that are pulse-width modulated(or modulated by a similar modulation technique including pulse-densitymodulation etc.). In pulse width modulation, an on-time of the switch 72is varied to determine the average current value output through thephase 36. In such modulation, the switches 72 are operated in switchedmode (either on or off) for energy efficiency, but switch at high ratesto produce continuous waveforms (e.g., sine waves of differentfrequencies) smoothed by the capacitance of the electrostatic motor 12In pulse width modulation, the switching speed of the semiconductors isat many times the fundamental frequency of the waveform of phases 36 andtypically more than 10-20 times that frequency.

Referring now to FIG. 5, an alternative version of the circuit shown inFIG. 4 breaks the inductor 70 into three inductor components 78 placedafter the connection of the phases 36 to the junctions between switchingswitches 72 of the H-bridge with one inductor component 78 placed in theseries with each of the phases 36. In this case, the voltage of thepower rails 74 and 76 feeding the H-bridge may be stabilized by acapacitor 80 extending between those rails.

Referring now to FIG. 6, a third topology of the current-source drive 40provides an advantage of allowing the switches 72 to operate at a lower,preferred voltage range than would be optimum for the voltage for thephases 36. In this circuit, precursor phases 82 are produced at thejunctions of each pair of switches 72 of the H-bridge (otherwiseidentical to the H-bridge of FIG. 4) and each of the precursor phases 82connect to a corresponding phase of a multi-phase primary 84 of a stepup transformer 86. The step up transformer 86 has a number of primaryphases equal to the number of phases 36 of the motor (three phases areshown in FIG. 6 comporting with a standard three phase motor).

The secondary winding 88 of the transformer 86 has an equal number oftaps as the input, but a greater number of turns, substantiallyincreasing the voltage applied to the phases 36 in comparison to thevoltage of the precursor phases 82. The step up in voltage may be atleast 3, but typically no more than 100.

Leakage flux from the transformer 86 provides the necessary inductivestorage of energy to promote current regulation in each of the phases36. In this embodiment, the output of each of the phases 36 may besensed by a current sensor 90 and at a voltage phase 92 so that separatecurrent and voltage measurements may be made and separate current andvoltage feedback loops may be used to control the switches 72 forimproved fidelity and stability. In this case, for example, currentfeedback may be used to provide d-q cross coupling decoupling.

Referring now to FIG. 7, as noted above, for a salient machine havingboth field and saliency torques, being an electrostatic motor 12 havingsignificant perturbations in rotor/stator capacitive coupling withrotation, the optimum angle of the input command signal may not bealigned along the q vector in d-q space but displaced slightly therefrombased on the construction of the motor. Such motors provide for twodifferent torque features including a primary field torque feature 102and a secondary elastance torque feature 104 each varying as a functionof the angle from the q-axis. The combination of these torques indicatedby curve 106 provides the net torque of the motor 12 and has beenmodeled by the present inventors to be maximized according to thefollowing formula:

$\gamma_{{ma}\; x} = {{{- a}\;{\sin\left( \frac{{C_{1{mf}}V_{fd}} \pm \sqrt{{C_{1{mf}}^{2}V_{fd}^{2}} + {8C_{2\; s}^{2}V_{s}^{2}}}}{4C_{2s}V_{s}} \right)}} \pm {5{\%.}}}$

where:

asin is the arcsin;

γ_(max) is an optimum angle magnitude between the desired d-q vector andthe q-axis of the command signal 34, normally a constant value forsteady-state motor operation;

C_(1mf) is the magnitude of the fundamental of the coupling capacitancebetween any one of the stator terminals (a, b, c) to any one of therotor terminals (Vf+ or Vf−);

C_(2s) is the magnitude of the second harmonic of the couplingcapacitance between any two stator terminals (a-b, b-c, c-a);

V_(fd) is the magnitude of the de rotor voltage; and

V_(s) is the magnitude of the fundamental of the stator voltage.

Because the values of C_(1mf), C_(2s), and V_(fd) typically vary as afunction of rotor position they are characterized by the frequencyspectrums of this periodic signal looking either at the fundamental orsecond harmonic as indicated. The value of V_(s) is normally, but neednot be, constant.

For motors that have only field torque, the value of γ_(max) may beselected from the group consisting of 0 and π (each ±5%) for the motorswith only saliency torque the value of γ_(max) may be selected from thegroup of π/4, −π/4, 3π/4, −3π/4 (each ±5%).

The term ABC is intended to represent an arbitrary number of phases notjust three phases as context would require.

Certain terminology is used herein for purposes of reference only, andthus is not intended to be limiting. For example, terms such as “upper”,“lower”, “above”, and “below” refer to directions in the drawings towhich reference is made. Terms such as “front”, “back”, “rear”, “bottom”and “side”, describe the orientation of portions of the component withina consistent but arbitrary frame of reference which is made clear byreference to the text and the associated drawings describing thecomponent under discussion. Such terminology may include the wordsspecifically mentioned above, derivatives thereof, and words of similarimport. Similarly, the terms “first”, “second” and other such numericalterms referring to structures do not imply a sequence or order unlessclearly indicated by the context. Although the stator and rotors areshown as disks in the disclosed embodiments, there is no requirementthat the stator or rotor be in a disk form.

When introducing elements or features of the present disclosure and theexemplary embodiments, the articles “a”, “an”, “the” and “said” areintended to mean that there are one or more of such elements orfeatures. The terms “comprising”, “including” and “having” are intendedto be inclusive and mean that there may be additional elements orfeatures other than those specifically noted. It is further to beunderstood that the method steps, processes, and operations describedherein are not to be construed as necessarily requiring theirperformance in the particular order discussed or illustrated, unlessspecifically identified as an order of performance. It is also to beunderstood that additional or alternative steps may be employed.

It is specifically intended that the present invention not be limited tothe embodiments and illustrations contained herein and the claims shouldbe understood to include modified forms of those embodiments includingportions of the embodiments and combinations of elements of differentembodiments as come within the scope of the following claims. All of thepublications described herein, including patents and non-patentpublications, are hereby incorporated herein by reference in theirentireties.

We claim:
 1. A variable speed drive for an electrostatic motor of a typehaving a stator with multiple stator electrodes adapted to generate arotating stator electric field vector about an axis and a rotor havingmultiple rotor electrodes providing a rotor electric field interactingwith the rotating electric field primarily by electrostatic forcesrather than magnetic forces, the variable speed drive comprising: a setof current-source drives adapted to connect to the multiple statorelectrodes wherein the current-source drives provide a set of electricalswitches in series with a current source implemented by an inductanceserving to regulate current to the stator electrodes; a rotor positiondetermination system; an ABC-dq transformation circuit receiving aposition signal from the rotor position detector and measures of outputsof the current-source drives to develop a measured d-q vector; an inputfor receiving the desired d-q vector; a comparison circuit providing anerror vector based on the measured d-q vector and desired d-q vector;and a dq-ABC transformation circuit receiving the position signal andproviding an output based on the error vector to develop a set ofelectrode signals provided to the current-source drives for driving thestator electrodes of the electrostatic motor.
 2. The variable speeddrive of claim 1 wherein the an inductance regulates current to thestator electrodes to within 25 percent of electrode signals provided tothe electrical switches for a motor of at least 10 watts.
 3. Thevariable speed drive of claim 2 wherein the current-source drivesinclude a multi-phase H-bridge of electrical switches receiving currentfrom an inductance operating to regulate current flow.
 4. The variablespeed drive of claim 2 wherein the current-source drive is a multiphaseH-bridge of electrical switches and includes current to threeinductances operating to regulate current flow to respective statorplates.
 5. The variable speed drive of claim 2 wherein thecurrent-source drive includes a multiphase H-bridge of electricalswitches providing current to a multiphase transformer system providinga current regulating inductance.
 6. The variable speed drive of claim 5wherein the current-source drive provides for a controller controllingswitching the electrical switches of the multiphase H-bridge accordingto a sensed current output and sensed voltage output of the multiphasetransformer.
 7. The variable speed drive of claim 1 further including aninput circuit generating the desired d-q vector based on a receivedcommand selected from the group of commands of torque and speed tocontrol current applied to the stator electrodes.
 8. The variable speeddrive of claim 7 wherein the input circuit selects an angle for thedesired d-q vector between the d- and q-motor axes as a function ofcoupling capacitance between a rotor terminal and a stator terminal anda coupling capacitance between two stator terminals.
 9. A variable speeddrive for an electrostatic motor of a type having a stator with multiplestator electrodes adapted to generate a rotating stator electric fieldvector about an axis and a rotor having multiple rotor electrodeproviding a rotor electric field interacting with the rotating electricfield primarily by electrostatic forces rather than magnetic forces, thevariable speed drive comprising: a set of current-source drives adaptedto connect to the multiple stator electrodes; a rotor positiondetermination system; an ABC-dq transformation circuit receiving aposition signal from the rotor position detector and measures of outputsof the current-source drives to develop a measured d-q vector; an inputfor receiving the desired d-vector; a comparison circuit providing anerror vector based on the measured d-q vector and desired d-q vector;and a dq-ABC transformation circuit receiving the position signal andproviding an output based on the error vector to develop a set ofelectrode signals provided to the current-source drives for driving thestator electrodes; wherein the input circuit selects an angle for thedesired d-q vector between the d- and q-motor axes according to aformula:$\gamma_{{ma}\; x} = {{{- a}\;{\sin\left( \frac{{C_{1{mf}}V_{fd}} \pm \sqrt{{C_{1{mf}}^{2}V_{fd}^{2}} + {8C_{2\; s}^{2}V_{s}^{2}}}}{4C_{2s}V_{s}} \right)}} \pm {5{\%.}}}$where: γ_(max) is an optimum angle magnitude between the desired d-qvoltage vector and the q-axis; C_(1mf) is a magnitude of a fundamentalof a coupling capacitance between any one of the stator terminals to anyone of the rotor terminals; C_(2s) is a magnitude of a second harmonicof a coupling capacitance between any two stator terminals; V_(fd) is amagnitude of the dc rotor voltage; and V_(s) is the magnitude of thefundamental of the stator voltage.
 10. The variable speed drive of claim1 wherein the input circuit selects an angle γ_(max) for the desired d-qvector between the d- and q-motor axes from the group consisting of 0and π (each ±5%).
 11. The variable speed drive of claim 1 wherein theinput circuit selects an angle γ_(max) for the desired d-q vectorbetween the d- and q-motor axes from the group consisting of π/4, −π/4,3π/4, −3π/4 (each ±5%).
 12. The variable speed drive of claim 1 whereinthe position detection system is selected from the group consisting of aposition encoder, an MMF sensor, and a saliency or MMF tracking sensorusing injected high-frequency current.
 13. The variable speed drive ofclaim 1 wherein the stator includes three electrically independent setsof electrodes in equal angles about the axis, the electrodes of each setjoined electrically to a common terminal.
 14. The variable speed driveof claim 1 further including an electrostatic motor of a type having astator with multiple stator electrodes receiving output from thecurrent-source drives and adapted to generate a rotating stator electricfield vector about the axis and a rotor having multiple rotor electrodesproviding a rotor electric field interacting with the rotating electricfield primarily by electrostatic forces rather than magnetic forces. 15.The variable speed drive of claim 14 further including an insulatingliquid contained to be present between the stator electrodes and rotorelectrodes.
 16. A method of providing variable speed control of anelectrostatic motor of a type having a stator with multiple statorelectrodes adapted to generate a rotating stator electric field vectorabout an axis and a rotor having multiple rotor electrodes providing arotor electric field interacting with the rotating electric fieldprimarily by electrostatic forces rather than magnetic forces,comprising the steps of: providing a set of current-source drives forproviding current to the stator electrodes wherein the current-sourcedrives provide a set of electrical switches in series with a currentsource implemented by an inductance serving to regulate current to thestator electrodes; receiving a position signal from the rotor positiondetector and measures of outputs of the current-source drives to developa measured d-q vector; receiving a desired d-q vector and comparing itto the measured d-q vector to produce an error vector; and transformingthe error vector to produce a set of outputs provided to thecurrent-source drives for driving the stator electrodes of theelectrostatic motor.
 17. The method of claim 16 further including thestep of generating the desired d-q current vector based on a receivedcommand selected from commands selected from the group consisting oftorque and speed to control current applied to the stator electrodes.18. The method of claim 17 wherein the desired d-q voltage vector has anangle between the d- and q-motor axes that is a function of couplingcapacitance between a rotor electrode and a stator electrode and acoupling capacitance between two stator electrodes.
 19. The method ofclaim 18 wherein the torque input circuit selects an angle for thedesired d-q voltage vector between the d- and q-motor axes according toa formula:$\gamma_{{ma}\; x} = {{- \frac{{C_{1{mf}}V_{fd}} \pm \sqrt{{C_{1{mf}}^{2}V_{fd}^{2}} + {8C_{2\; s}^{2}V_{s}^{2}}}}{4C_{2s}V_{s}}} \pm {5\%}}$where: γ_(max) is an optimum angle between the desired d-q vector andthe q-axis; C_(1mf) is a magnitude of a fundamental of a couplingcapacitance between any one of the stator terminals to any one of therotor terminals; C_(2s) is a magnitude of a second harmonic of acoupling capacitance between any two stator terminals; V_(fd) is amagnitude of a dc rotor voltage; and V_(s) is the magnitude of thefundamental of the stator voltage.
 20. The method of claim 16 whereinthe position detection system is selected from the group consisting of aposition encoder, an MMF sensor, and a saliency or MMF tracking sensorusing injected high-frequency current.